Long division unites - long union divides, a model for social network evolution

A remarkable phenomenon in the time evolution of many networks such as cultural, political, national and economic systems, is the recurrent transition between the states of union and division of nodes. In this work, we propose a phenomenological modeling, inspired by the maxim “long union divides and long division unites”, in order to investigate the evolutionary characters of these networks composed of the entities whose behaviors are dominated by these two events. The nodes are endowed with quantities such as identity, ingredient, richness (power), openness (connections), age, distance, interaction etc. which determine collectively the evolution in a probabilistic way. Depending on a tunable parameter, the time evolution of this model is mainly an alternative domination of union or division state, with a possible state of final union dominated by one single node.

💡 Research Summary

The paper presents a phenomenological model that captures the recurrent alternation between unification (union) and fragmentation (division) observed in many large‑scale social, cultural, political, and economic networks. Each node is endowed with a set of attributes: a five‑dimensional identity vector, a “richness” representing development or power, an age, a degree (number of connections), and an “ingredient” I that counts how many previous nodes have merged into it. Identity distance is Euclidean, richness grows linearly with age and ingredient, and degree reflects openness to information exchange.

Merging occurs between two adjacent nodes i and j with probability

p_m(i,j) = A·(k_i + k_j)·(Δr_{ij}+1) / d_{ij},

where k denotes degree, Δr the difference in richness, d the identity distance, and A a normalization constant. This formulation embodies the intuition that highly connected, similarly rich, and culturally close entities are more likely to unite. When a merge happens, a new composite node n is created with:

• ingredient I_n = I_i + I_j (conserved total ingredient),
• age a_n = 1,
• richness r_n = r_i + r_j,
• identity = component‑wise average of the parents,
• degree k_n = k_i + k_j – 2 – N_common, where N_common is the number of shared neighbors.

Splitting applies only to composite nodes (I > 1). A composite node with ingredient I can split into I new nodes with probability

p_s = (1/Z_s)·I^a·(k + k_c)·r,

where a is the node’s age, k its degree, r its richness, k_c a tunable parameter, and Z_s a normalization factor. Larger ingredient, older age, and higher degree increase the chance of division, while higher richness suppresses it. In the simulations k_c is fixed to 1, ensuring that even isolated composites have a non‑zero chance to split. After splitting, each offspring inherits the parent’s richness, receives a randomly generated identity vector, and is linked to each other and to the former neighbors with a constant probability p_rc = 0.09.

The simulation starts with N = 2000 single‑ingredient nodes randomly connected with probability p_c = 0.3. All nodes initially have age, richness, and ingredient equal to 1. At each time step a node is selected uniformly at random. If its ingredient I = 1, only merging is possible; if I > 1, merging or splitting is chosen with equal probability. Nodes that have just been created (either by merging or splitting) are excluded from being selected in the next step, preventing immediate cascades.

Results reveal a striking cyclical dynamics. Network size N(t) and average degree ⟨k⟩ oscillate: an initial phase of rapid merging reduces the network to a single giant node (N = 1), after which stochastic splitting generates new nodes and the cycle repeats. This alternation directly mirrors the proverb “long union divides, long division unites.” Degree distributions evolve from an initial Poisson‑like peak (random graph) to a bimodal shape during merging (one peak for high‑degree composite nodes, another for low‑degree neighbors), and later collapse back to a single sharp peak as the system approaches the end of a cycle. The shift direction depends on whether merging or splitting dominates.

Ingredient distributions display a transition between power‑law‑like heavy tails and near‑uniform shapes. Early in a merging‑dominated phase, most nodes have small I while a few acquire very large I, producing a fat‑tailed distribution. As merging continues, the distribution flattens, becoming almost uniform. When splitting becomes prevalent, the large‑I nodes fragment, restoring a power‑law tail. This back‑and‑forth transition repeats throughout the simulation, illustrating how the system continuously reshapes the hierarchy of composite entities.

Node lifetimes (the duration a node exists before being merged or split) also show a mixed pattern: many short‑lived nodes coexist with a few long‑lived composites, reflecting the stochastic balance of the two processes.

The authors discuss the model’s strengths—its incorporation of multiple realistic attributes and its ability to reproduce empirically observed cycles of unification and fragmentation—and its limitations, such as sensitivity to parameter choices, lack of calibration against real‑world data, and omission of asymmetric power structures or external shocks (e.g., wars, natural disasters). They suggest future work should include empirical validation, systematic parameter sensitivity analysis, incorporation of exogenous events, and extension to multilayer or hierarchical networks.

In conclusion, the paper provides a mathematically tractable framework that captures the essence of the “long union divides, long division unites” phenomenon, offering insights into how complex social systems may self‑organize through alternating phases of aggregation and disaggregation. This contributes a useful theoretical tool for scholars and policymakers interested in the dynamics of large‑scale collective entities.